Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimpnfxnegmnf2 | Structured version Visualization version GIF version |
Description: A sequence converges to +∞ if and only if its negation converges to -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
Ref | Expression |
---|---|
xlimpnfxnegmnf2.j | ⊢ Ⅎ𝑗𝐹 |
xlimpnfxnegmnf2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
xlimpnfxnegmnf2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimpnfxnegmnf2.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
Ref | Expression |
---|---|
xlimpnfxnegmnf2 | ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimpnfxnegmnf2.j | . . 3 ⊢ Ⅎ𝑗𝐹 | |
2 | xlimpnfxnegmnf2.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | xlimpnfxnegmnf2.f | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
4 | 1, 2, 3 | xlimpnfxnegmnf 42115 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥)) |
5 | xlimpnfxnegmnf2.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | 1, 5, 2, 3 | xlimpnf 42143 | . 2 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))) |
7 | nfmpt1 5164 | . . . 4 ⊢ Ⅎ𝑗(𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗)) | |
8 | 3 | ffvelrnda 6851 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ*) |
9 | 8 | xnegcld 12694 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -𝑒(𝐹‘𝑘) ∈ ℝ*) |
10 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑘-𝑒(𝐹‘𝑗) | |
11 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑗𝑘 | |
12 | 1, 11 | nffv 6680 | . . . . . . 7 ⊢ Ⅎ𝑗(𝐹‘𝑘) |
13 | 12 | nfxneg 41757 | . . . . . 6 ⊢ Ⅎ𝑗-𝑒(𝐹‘𝑘) |
14 | fveq2 6670 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘)) | |
15 | 14 | xnegeqd 41731 | . . . . . 6 ⊢ (𝑗 = 𝑘 → -𝑒(𝐹‘𝑗) = -𝑒(𝐹‘𝑘)) |
16 | 10, 13, 15 | cbvmpt 5167 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗)) = (𝑘 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑘)) |
17 | 9, 16 | fmptd 6878 | . . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗)):𝑍⟶ℝ*) |
18 | 7, 5, 2, 17 | xlimmnf 42142 | . . 3 ⊢ (𝜑 → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥)) |
19 | 2 | uztrn2 12263 | . . . . . . 7 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍) |
20 | xnegex 12602 | . . . . . . . . 9 ⊢ -𝑒(𝐹‘𝑗) ∈ V | |
21 | fvmpt4 41528 | . . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑍 ∧ -𝑒(𝐹‘𝑗) ∈ V) → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) = -𝑒(𝐹‘𝑗)) | |
22 | 20, 21 | mpan2 689 | . . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) = -𝑒(𝐹‘𝑗)) |
23 | 22 | breq1d 5076 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ -𝑒(𝐹‘𝑗) ≤ 𝑥)) |
24 | 19, 23 | syl 17 | . . . . . 6 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ -𝑒(𝐹‘𝑗) ≤ 𝑥)) |
25 | 24 | ralbidva 3196 | . . . . 5 ⊢ (𝑘 ∈ 𝑍 → (∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥)) |
26 | 25 | rexbiia 3246 | . . . 4 ⊢ (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥) |
27 | 26 | ralbii 3165 | . . 3 ⊢ (∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))‘𝑗) ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥) |
28 | 18, 27 | syl6bb 289 | . 2 ⊢ (𝜑 → ((𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)-𝑒(𝐹‘𝑗) ≤ 𝑥)) |
29 | 4, 6, 28 | 3bitr4d 313 | 1 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (𝑗 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑗))~~>*-∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2961 ∀wral 3138 ∃wrex 3139 Vcvv 3494 class class class wbr 5066 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 ℝcr 10536 +∞cpnf 10672 -∞cmnf 10673 ℝ*cxr 10674 ≤ cle 10676 ℤcz 11982 ℤ≥cuz 12244 -𝑒cxne 12505 ~~>*clsxlim 42119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-z 11983 df-uz 12245 df-xneg 12508 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-topgen 16717 df-ordt 16774 df-ps 17810 df-tsr 17811 df-top 21502 df-topon 21519 df-bases 21554 df-lm 21837 df-xlim 42120 |
This theorem is referenced by: (None) |
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